A tunable inductor, as described herein, is preferably used in so called “lumped elements”, which use discrete inductors and capacitors. Accordingly, preferred implementations of a tunable resonator include mechanically tunable resonators. In some implementations, for instance, a tunable resonator uses the lumped tunable capacitor and lumped tunable inductors described in U.S. patent application Ser. No. 13/848,682, filed Mar. 21, 2013, now U.S. Pat No. 9,048,023, issued Jun. 2, 2015 and entitled “TUNABLE CAPACITOR” and U.S. patent application Ser. No. 13/848,692, filed Mar. 21, 2013 and entitled “TUNABLE INDUCTOR,” now abandoned, the contents of which are incorporated herein for all purposes.
Conventional resonators are made up of an inductor and a capacitor. They can create a series circuit or shunt circuit as shown in FIG. 1. The resonant frequency of such resonator is:
                                          f            O                    =                      1                          2              ⁢                              π                ·                                                      L                    ·                    C                                                                                      ,                                  ⁢                  where          ⁢                                          ⁢                      f            0                    ⁢                                          ⁢          is          ⁢                                          ⁢          a          ⁢                                          ⁢          frequency          ⁢                                          ⁢          of          ⁢                                          ⁢          free          ⁢                                          ⁢          oscillation                ,                                  ⁢                  L          ⁢                                          ⁢          is          ⁢                                          ⁢          inductance          ⁢                                          ⁢          value          ⁢                                          ⁢          of          ⁢                                          ⁢          the          ⁢                                          ⁢          inductor                ,                                  ⁢                  C          ⁢                                          ⁢          is          ⁢                                          ⁢          capacitance          ⁢                                          ⁢          value          ⁢                                          ⁢          of          ⁢                                          ⁢          the          ⁢                                          ⁢                      capacitor            .                                              (        1        )            
The prior art of the tunable resonator uses either a tunable inductor or a tunable capacitor, but never both. As seen from formula (1), when inductance or capacitance of the resonator changes, the resonant frequency changes as well.
Certain applications of a tunable resonator, such as a tunable filter, for example, require the resonator to have a constant bandwidth even when central frequency of the filter changes. In order to keep the bandwidth constant, the resonators must be able to change their resonant frequency while keeping a characteristic impedance constant. The resonator's characteristic impedance can be expressed as:
                                          Z            o                    =                                    L              C                                      ,                            (        2        )            
Formulas (1) and (2) demonstrate that when only one parameter changes, either L or C, the central frequency changes, yet the impedance changes as well. This is a main weakness of the prior art of tunable resonators. The change of impedance results in distortion the frequency response curve and change of the bandwidth when tuned over a frequency range, as shown in FIG. 2. This limits the tunable frequency range to a narrow band where the distortion could be tolerated.